WORK AND ENERGY

104. Work.?"Whenever a force moves a body upon which it
acts, it is said to do work upon that body." For example, if a man
pushes a wheelbarrow along a path, he is doing work on it as long as the
wheelbarrow moves, but if the wheelbarrow strikes a stone and the man
continues to push and no motion results, from a scientific point of view
he is then doing no work on it.
"Work signifies the overcoming of resistance," and unless the
resistance is overcome no work is done. Lifting a weight is doing work
on it, supporting a weight is not, although the latter may be nearly as
tiresome as the former. Work as used in science is a technical term. Do
not attach to it meanings which it has in every-day speech.105. Measurement of Work.?Work is measured by the product of the force by the displacement caused in the direction of the force, that is W = fs.
Therefore if a unit of force acts through a unit of space, a unit of
work will be done. There are naturally several units of work depending
upon the units of force and space employed.English Work Unit.?If the force of one pound acts through the distance of one foot, a foot-pound of work is done. A foot-pound is defined as the work done when 1 lb. is lifted 1 ft. against the force of gravity.Metric Work Unit.?If the force is one kilogram and the distance one meter, one kilogram-meter of work is done.Absolute Work Unit.?If the force of one dyne acts through the distance of one centimeter a dyne-centimeter[Pg 120]
of work is done. This usually is called an i. Other work units are
sometimes used depending upon the force and distance units employed.
One, the i, is equal to 10,000,000 ergs or 107 ergs.

Problem.?If a load is drawn 2 miles by a team exerting 500 lbs. force, how much work is done?Solution.?Since the force employed is 500 lbs., and the distance is 2 ? 5280 ft., the work done is 500 ? 2 ? 5280 or 5,280,000 ft.-lbs.

106. Energy.?In the various cases suggested in the
paragraphs upon work, an agent, a man, an animal or a machine, was
mentioned as putting forth an effort in order to do the work. It is also
true that in order to perform work an agent must employ energy, or the energy of a body is its capacity for doing work.
Where an agent does work upon a body, as in winding up a spring or in
lifting a weight, the body upon which the work has been done may acquire
energy by having work done upon it. That is, it may become able to do
work itself upon some other body. For instance, a lifted weight in
falling back to its first position may turn wheels, or drive a post into
the ground against resistance; a coiled spring may run clock work,
strike a blow, or close a door. Hence the energy, or the capacity for
doing work, is often acquired by a body because work has first been done
upon that body.107. Potential Energy.?The wound up spring may do work
because work has first been done upon it. The lifted weight may also do
work because work has first been done in raising it to its elevated
position since in falling it may grind an object to powder, lift another
weight or do some other kind of work. The energy that a body
possesses on account of its position or shape and a stress to which it
is subjected is called potential energy. The potential energy of a body is measured by the work done in lifting it,[Pg 121]
changing its shape, or by bringing about the conditions by which it can
do work. Thus if a block of iron weighing 2000 lbs. is lifted 20 ft.,
it possesses 40,000 ft.-lbs. of potential energy. It is therefore able
to do 40,000 ft.-lbs. of work in falling back to its first position. If
the block just mentioned should fall from its elevated position upon a
post, it could drive the post into the ground because its motion at the
instant of striking enables it to do work. To compute potential energy
you compute the work done upon the body. That is, P.E. = w ? h or f ? s.108. Kinetic Energy.?The energy due to the motion of a body is called kinetic energy.
The amount of kinetic energy in a body may be measured by the amount of
work done to put it in motion. It is usually computed, however, by
using its mass and velocity on striking. To illustrate, a 100-lb. ball
is lifted 16 ft. The work done upon it, and hence its potential energy,
is 1600 ft.-lbs. On falling to the ground again, this will be changed
into kinetic energy, or there will be 1600 ft.-lbs. of kinetic energy on
striking. It will be noted that since energy is measured by the work it
can do, work units are always used in measuring energy. To compute the
kinetic energy of a falling body by simply using its mass and velocity
one proceeds as follows, in solving the above problem:

First, find the velocity of the falling body which has fallen 16 ft. A body falls 16 ft. in one second. In this time it gains a velocity of 32 ft. per second. Now using the formula for kinetic energy K.E. = wv2/(2g), we have K.E. = 100 ? 32 ? 32/(2 ? 32) = 1600 ft.-lbs. as before. The formula, K.E. = wv2/(2g), may be derived in the following manner:
The kinetic energy of a falling body equals the work done in giving it its motion, that is, K.E. = w ? S, in which, w = the weight of the body and S = the distance the body must fall freely in order [Pg 122]to acquire its velocity. The distance fallen by a freely falling body, S, = 1/2gt2 = g2t2/(2g) (Art. 98, p. 111). Now, v = gt and v2 = g2t2.
Substituting for g2t2, its equal v2, we have S = v2/(2g). Substituting this value of S in the equation K.E. = w ? S, we have K.E. = wv2/(2g).
Since the kinetic energy of a moving body depends upon its mass and velocity and not upon the direction
of motion, this formula may be used to find the kinetic energy of any
moving body. Mass and weight in such problems may be considered
numerically equal.

Important Topics
1. Work defined.
2. Work units, foot-pound, kilogram-meter, erg.
3. Energy defined.
4. Kinds of energy, potential and kinetic.Problems
1. How much work will a 120-lb. boy do climbing a mountain 3000 ft. high? Should the vertical or slant height be used? Why?
2. In a mine 4000 kg. of coal are lifted 223 meters: how much work
is done upon the coal? What is the kind and amount of energy possessed
by the coal?
3. A pile driver weighs 450 lbs. It is lifted 16 ft. How much work
has been done upon it? What kind and amount of energy will it have after
falling 16 ft. to the pile?
4. A train weighing 400 tons is moving 30 miles per hour. Compute
its kinetic energy. (Change its weight to pounds and velocity to feet
per second.)
5. What would be the kinetic energy of the train in problem 4 if it
were going 60 miles per hour? If it were going 90 miles per hour? How
does doubling or trebling the speed of an object affect its kinetic
energy? How does it affect its momentum?
6. What is the kinetic energy of a 1600-lb. cannon ball moving 2000 ft. per second?
7. Mention as many kinds of mechanical work as you can and show how each satisfies the definition of work.
8. A pile driver weighing 3000 lbs. is lifted 10 ft. How much work is done upon it?[Pg 123]
9. If the pile driver in problem 8 is dropped upon the head of a
pile which meets an average resistance of 30,000 lbs., how far will one
blow drive it?
10. A 40 kg. stone is placed upon the top of a chimney 50 meters high. Compute the work done in kilogram-meters and foot-pounds.
(2) Power and Energy109. Horse-power.?In computing work, no account is taken of
the time required to accomplish it. But since the time needed to perform
an undertaking is of much importance, the rate of work, or the power or activity
of an agent is an important factor. Thus if one machine can do a piece
of work in one-fifth the time required by another machine, it is said to
have five times the power of the other. Therefore the power of a
machine is the rate at which it can do work. James Watt (1736-1819), the inventor of the steam-engine, in expressing the power of his engine, used as a unit a horse-power.
He considered that a horse could do 33,000 ft.-lbs. of work a minute.
This is equal to 550 ft.-lbs. per second or 76.05 kg.-m. per second.
This is too high a value but it has been used ever since his time. Steam
engines usually have their power rated in horse-power. That is,
locomotives produce from 500 to 1500 horse-power. Some stationary and
marine engines develop as high as 25,000 horse-power. The power of an
average horse is about 3/4 horse-power and of a man about 1/7
horse-power when working continuously for several hours.110. The Watt.?In the metric system, the erg as a unit of
work would give as a unit of power 1 erg per second. This amount is so
small, however, that a larger unit is usually employed, the practical
unit being 10,000,000 ergs a second, that is, one joule per second. (See
Art. 105.) This practical unit is called a Watt after James Watt.[Pg 124]
The power of dynamos is usually expressed in kilowatts, a kilowatt
representing 1000 watts. Steam-engines in modern practice are often
rated in kilowatts instead of horse-power. A horse-power is equivalent
to 746 watts, or is nearly 3/4 of a kilowatt.111. Energy. Its Transference and Transformation. We have
considered energy as the capacity for doing work, and noted the two
kinds, potential and kinetic, and the facility with which one may change
into another. In fact, the transference of energy from one body to
another, and its transformation from one form to another is one of the
most common processes in nature. Take a pendulum in motion, at the end of a swing, its energy being entirely due to its elevated position is all potential; at the lowest point in its path its energy being entirely due to its motion is all kinetic.
The change goes on automatically as long as the pendulum swings. A
motor attached by a belt to a washing machine is started running. The
energy of the motor is transferred by the belt to the washer where it is
used in rubbing and moving the clothes.
The heat used in warming a house is usually obtained by burning
coal or wood. Coal is believed to be formed from the remains of plants
that grew in former geologic times. These plants grew through the help
of the radiant energy of the sun. The following are transformations of
energy that have occurred: The radiant energy of sunlight was
transformed into the chemical energy of the plants. This remained
as chemical energy while the plants were being converted into coal, was
mined, brought to the stove or furnace and burned. The burning
transformed the chemical energy into heat energy in which form we use it
for warming rooms. Take the energy used in running a street car whose
electrical energy comes from a waterfall. The energy of the car itself
is mechanical.[Pg 125]
Its motor, however, receives electrical energy and transforms it into
mechanical. This electrical energy comes along a wire from a dynamo at
the waterfall, where water-wheels and generators transform into
electrical energy the mechanical energy of the falling water. The water
obtained its energy of position by being evaporated by the heat of the
radiant energy of the sun. The vapor rising into the air is condensed
into clouds and rain, and falling on the mountain side, has, from its
elevated position, potential energy. The order of transformation,
therefore, is in this case, radiant, heat, mechanical, electrical, and
mechanical. Can you trace the energy from the sun step by step to the
energy you are using in reading this page?112. Forms of Energy.?A steam-engine attached to a train of cars employs its energy in setting the cars in motion, i.e.,
in giving them kinetic energy and in overcoming resistance to motion.
But what is the source of the energy of the engine? It is found in the
coal which it carries in its tender. But of what kind? Surely not
kinetic, as no motion is seen. It is therefore potential. What is the
source of the energy of the coal? This question leads us back to the
time of the formation of coal beds, when plants grew in the sunlight and
stored up the energy of the sun's heat and light as chemical
energy. The sun's light brings to the earth the energy of the sun, that
central storehouse of energy, which has supplied nearly all the
available energy upon the earth. Five forms of energy are known, viz., mechanical, heat, electrical, radiant, and chemical.113. Energy Recognized by its Effects.?Like force, energy is invisible and we are aware of the forms only by the effects produced by it.
We recognize heat by warming, by expansion, by pressure.[Pg 126]
We recognize light by warming, by its affecting vision.
We recognize electrical energy by its heat, light, motion, or magnetic effect. We recognize mechanical energy by the motion that it produces. We recognize chemical energy by knowing that the source of energy does not belong to any of the foregoing.
A boy or girl is able to do considerable work. They therefore
possess energy. In what form does the energy of the body mainly occur?
One can determine this for himself by applying questions to each form of
energy in turn as in Art. 114.114. Source of the Energy of the Human Body.?Is the energy
of the human body mostly heat? No, since we are not very warm. Is it
light or electrical? Evidently not since we are neither luminous nor
electrical. Is it mechanical? No, since we have our energy even when at
rest. Is it chemical? It must be since it is none of the others.
Chemical energy is contained within the molecule.
It is a form of potential energy and it is believed to be due to
the position of the atoms within the molecule. As a tightly coiled watch
spring may have much energy within it, which is set free on allowing
the spring to uncoil, so the chemical energy is released on starting the
chemical reaction. Gunpowder and dynamite are examples of
substances containing chemical energy. On exploding these, heat, light,
and motion are produced. Gasoline, kerosene, and illuminating gas are
purchased because of the potential energy they contain. This energy is
set free by burning or exploding them.
The source of the energy of our bodies is of course the food we
eat. The energy contained in the food is also chemical. Vegetables
obtain their energy from the sunlight[Pg 127]
(radiant energy). This is why plants will not grow in the dark. The
available energy is mostly contained in the form of starch, sugar and
oil. Digestion is employed principally to dissolve these substances so
that the blood may absorb them and carry them to the tissues of the body
where they are needed. The energy is set free by oxidation (burning),
the oxygen needed for this being supplied by breathing. Breathing also
removes the carbon dioxide, which results from the combustion. It is for
its energy that our food is mostly required.115. Conservation of Energy.?In the study of matter we
learned that it is indestructible. Energy is also believed to be
indestructible. This principle stated concisely teaches that despite the innumerable changes which energy undergoes the amount in the universe is unchangeable,
and while energy may leave the earth and be lost as far as we are
concerned, that it exists somewhere in some form. The principle which
teaches this is called the "Conservation of Energy." The form into which
energy is finally transformed is believed to be heat.

Exercises

1. A boy weighing 110 lbs. ran up a stairs 10 ft. high, in 4 seconds. How much work was done? What was his rate of work (foot-pounds per second)? Express also in horse-power.
2. A locomotive drawing a train exerts a draw bar pull of 11,000 lbs. How much work does it do in moving 3 miles? What[Pg 128] is its rate of work if it moves 3 miles in 5 minutes? Express in horse-power.[G]
3. If 400 kg. are lifted 35 meters in 5 seconds what work is done?
What is the rate of work? Express in horse-power, watts and kilowatts.
4. Trace the energy of a moving railway train back to its source in the sun.
5. Why does turning the propeller of a motor boat cause the boat to move?
6. Does it require more power to go up a flight of stairs in 5
seconds than in 10 seconds? Explain. Is more work done in one case than
in the other? Why?
7. Can 1 man carrying bricks up to a certain elevation for 120 days do as much work as 120 men carrying up bricks for 1 day?
8. If the 1 man and 120 men of problem 7 do the same amount of work have they the same power? Explain.
9. If 160 cu. ft. of water flow each second over a dam 15ft. high what is the available power?
10. What power must an engine have to fill a tank 11 ? 8 ? 5 ft. with water 120 ft. above the supply, in 5 minutes?
11. A hod carrier weighing 150 lbs. carries a load of bricks
weighing 100 lbs. up a ladder 30 ft. high. How much work does he do?
12. How much work can a 4-horse-power engine do in 5 minutes?
13. Find the horse-power of a windmill that pumps 6 tons of water from a well 90 ft. deep in 30 minutes.
14. How many horse-power are there in a waterfall 20 ft. high over which 500 cu. ft. of water pass in a minute?
15. The Chicago drainage canal has a flow of about 6000 cu. ft. a
second. If at the controlling works there is an available fall of 34 ft.
how many horse-power can be developed?
16. How long will it take a 10-horse-power pump to fill a tank of 4000 gallons capacity, standing 300 ft. above the pump?
17. A boy weighing 162 lbs. climbs a stairway a vertical height of 14 ft. in 14.6 seconds. How much power does he exert?
18. The same boy does the same work a second time in 4.2 seconds.
How much power does he exert this time? What causes the difference?
19. What is a horse-power-hour? a kilowatt-hour?[Pg 129]
(3) Simple Machines and the Lever116. Machines and Their Uses.?A man, while standing on the ground, can draw a flag to the top of a pole, by using a rope passing over a pulley.
A boy can unscrew a tightly fitting nut that he cannot move with his fingers, by using a wrench.
A woman can sew a long seam by using a sewing machine in much less time than by hand.
A girl can button her shoes much quicker and easier with a button-hook than with her fingers.
These illustrations show some of the reasons why machines are used.
In fact it is almost impossible to do any kind of work efficiently
without using one or more machines.117. Advantages of Machines.?(a) Many machines make possible an increased speed as in a sewing machine or a bicycle.
(b) Other machines exert an increased force. A rope and a
set of pulleys may enable a man to lift a heavy object such as a safe or
a piano. By the use of a bar a man can more easily move a large rock.
(See Fig. 83.)

Fig. 83.?The rock is easily moved.

(c) The direction of a force may be changed thus enabling
work to be done that could not be readily accomplished otherwise. As,
e.g., the use of a pulley in raising a flag to the top of a flag pole,
or in raising a bucket of ore from a mine by using a horse attached to a
rope passing over two or more pulleys. (See Fig. 84.)
(d) Other agents than man or animals can be used such as electricity, water power, the wind, steam, etc. Fig. 85 represents a windmill often used in pumping water.[Pg 130]A machine is a device for transferring or transforming energy. It is usually therefore an instrument for doing work. An electric motor is a machine since it transforms the energy of the electric current into motion or mechanical energy, and transfers the energy from the wire to the driving pulley.

Fig. 84.?The horse lifts the bucket of ore.

118. A Machine Cannot Create Energy.?Whatever does work upon
a machine (a man, moving water, wind, etc.) loses energy which is
employed in doing the work of the machine. A pair of shears is a machine
since it transfers energy from the hand to the edges that do the
cutting. Our own bodies are often considered as machines since they both
transfer and transform energy.
We must keep in mind that a machine cannot create energy.
The principle of "Conservation of Energy" is just as explicit on one
side as the other. Just as energy, cannot be destroyed, so energy cannot
be created. A machine can give out no more energy than is given to it.
It acts simply as an agent in transferring energy from one[Pg 131]
body to another. Many efforts have been made to construct machines that
when once started will run themselves, giving out more energy than they
receive. Such efforts, called seeking for perpetual motion, have never succeeded. This fact is strong evidence in favor of the principle of the conservation of energy.

Fig. 85.?A windmill.

119. Law of Machines.?When a body receives energy, work is
done upon it. Therefore work is done upon a machine when it receives
energy and the machine does work upon the body to which it gives the
energy. In the operation of a machine, therefore, two quantities of work
are to be considered and by the principle of the conservation of
energy, these two must be equal. The work done by a machine equals the work done upon it, or the energy given out by a machine equals the energy received by it. These two quantities of work must each be composed of a force factor and a space
factor. Therefore two forces and two spaces are to be considered in the
operation of a machine. The force factor of the work done on the
machine is called the force or effort. It is the force applied to the machine. The force factor of the work done by a machine is called the weight or resistance. It is the force exerted by the machine in overcoming the resistance and equals the resistance overcome.[Pg 132]
If f represents the force or effort, and Df the space it acts through, and w represents the weight or resistance, and Dw the space it acts through, then the law of machines may be expressed by an equation, f ? Df = w ? Dw. That is, the effort times the distance the effort acts equals the resistance times the distance the resistance is moved or overcome.
When the product of two numbers equals the product of two other numbers
either pair may be made the means and the other the extremes of a
proportion. The equation given above may therefore be expressed w: f = Df: Dw.
Or the resistance is to the effort as the effort distance is to the
resistance distance. The law of machines may therefore be expressed in
several ways. One should keep in mind, however, that the same law of machines is expressed even though the form be different. What two ways of expressing the law are given?120. The Simple Machines.?There are but six simple machines.
All the varieties of machines known are simply modifications and
combinations of the six simple machines. The six simple machines are
more easily remembered if we separate them into two groups of three
each. The first or lever group consists of those machines in which a part revolves about a fixed axis. It contains the lever, pulley and wheel and axle. The second or inclined plane group includes those having a sloping surface. It contains the inclined plane, the wedge, and the screw.121. The Lever.?The lever is one of the simple machines most frequently used, being seen in scissors, broom, coal shovel, whip, wheelbarrow, tongs, etc. The lever consists of a rigid bar capable of turning about a fixed axis called the fulcrum.
In studying a lever, one wishes to know what weight or resistance it
can overcome when a certain force is applied to it. Diagrams of levers,
therefore, contain the letters w and f. In addition to these, O[Pg 133] stands for the fulcrum on which it turns. By referring to Fig. 86, a, b, c,
one may notice that each of these may occupy the middle position
between the other two. The two forces (other than the one exerted by the
fulcrum) acting on a lever always oppose each other in the matter of
changing rotation. They may be considered as a pair of parallel forces
acting on a body, each tending to produce rotation.

Fig. 86.?The three classes of levers.

122. Moment of Force.?The effectiveness of each force may therefore be determined by computing its moment
about the fixed axis (see Art. 84), that is, by multiplying each force
by its distance to the fulcrum or axis of rotation. Let a meter stick
have a small hole bored through it at the 50 cm. mark near one edge, and
let it be mounted on a nail driven into a vertical support and balanced
by sliding a bent wire along it. Suspend by a fine wire or thread a 100
g. weight, 15 cm. from the nail and a 50 g. weight 30 cm. from the
nail, on the other side of the support. These two weights will be found
to balance. When viewed from this side A (Fig. 87) tends to turn the lever in a clockwise direction (down at right), B
in the counter-clockwise direction (down at left). Since the lever
balances, the forces have equal and opposite effects in changing its
rotation as may also be computed by determining the[Pg 134] moment of each force by multiplying each by its distance from the fulcrum. Therefore the effectiveness of a force in changing rotation depends upon the distance from it to the axis as well as upon the magnitude of the force.

Fig. 87.?The two moments are equal about C. 100 ? 15 = 50 ? 30

From the experiment just described, the moment of the acting force equals the moment of the weight or f ? Df = w ? Dw,
or the effort times the effort arm equals the weight times the weight
arm. This equation is called the law of the lever. It corresponds to the
general law of machines and may also be written w: f = Df: Dw.123. Mechanical Advantage.?A lever often gives an advantage
because by its use one may lift a stone or weight which the unaided
strength of man could not move. If the lever is used in lifting a stone
weighing 500 lbs., the force available being only 100 lbs., then its mechanical advantage would be 5, the ratio of w:f.
In a similar way, the mechanical advantage of any machine is found by
finding the ratio of the resistance or weight to the effort. What must
be the relative lengths of the effort arm and resistance or weight arm
in the example just mentioned? Since the effort times the effort arm
equals the weight times the weight arm, if f ? Df = w ? Dw, then Df is five times Dw. Hence the mechanical advantage of a lever is easily found by finding the ratio of the effort arm to the weight arm.

Important Topics

Exercises

1. Give six examples of levers you use.
2. Fig. 88a represents a pair of paper shears, 88b a
pair of tinner's shears. Which has the greater mechanical advantage?
Why? Explain why each has the most effective shape for its particular
work.

Fig. 88.?(a) Paper shears. (b) Tinner's shears.

3. Find examples of levers in a sewing machine.
4. What would result if, in Art. 122, the 100 g. weight were put 25
cm. from O and the 50 g. weight 45 cm. from O? Why? Explain using
principle of moments.
5. How is the lever principle applied in rowing a boat?
6. When you cut cardboard with shears, why do you open them wide and cut near the pivot?
7. In carrying a load on a stick over the shoulder should the pack be carried near the shoulder or out on the stick? Why?
8. How can two boys on a see-saw start it without touching the ground?
9. In lifting a shovel full of sand do you lift up with one hand as hard as you push down with the other? Why?

Fig. 89.?The hammer is a bent lever. What is its mechanical advantage?

10. Why must the hinges of a gate 3 ft. high and 16 ft. wide be stronger than the hinges of a gate 16 ft. high and 3 ft. wide?
11. When one sweeps with a broom do the hands do equal amounts of work? Explain.[Pg 136]
12. A bar 6 ft. long is used as a lever to lift a weight of 500
lbs. If the fulcrum is placed 6 in. from the weight, what will be the
effort required? Note: two arrangements of weight, fulcrum and effort
are possible.
13. The handle of a hammer is 12 in. long and the claw that is used
in drawing a nail is 2.5 in. long. (See Fig. 89.) A force of 25 lbs. is
required to draw the nail. What is the resistance of the nail?
14. The effective length of the head of a hammer is 2 in. The
handle is 15 in. long and the nail holds in the wood with a force of 500
lbs. Only 60 lbs. of force is available at the end of the handle. What
will be the result?
15. If an effort of 50 lbs. acting on a machine moves 10 ft., how far can it lift a weight of 1000 lbs.?
16. A bar 10 ft. long is to be used as a lever. The weight is kept 2
ft. from the fulcrum. What different levers can it represent?
17. The effort arm of a lever is 6 ft., the weight arm 6 in. How long will the lever be? Give all possible answers.
18. Two boys carry a weight of 100 lbs. on a pole 5 ft. long
between them. Where should the weight be placed in order that one boy
may carry one and one-fourth times as much as the other?

(4) The Wheel and Axle and the Pulley

124. The Wheel and Axle.?1. One of the simple machines most commonly applied in compound machines is the wheel and axle. It consists of a wheel H mounted on a cylinder Y
so fastened together that both turn on the same axis. In Fig. 90, ropes
are shown attached to the circumferences of the wheel and axle.
Sometimes a hand wheel is used as on the brake of a freight or street
car, or simply a crank and handle is used, as in Fig. 91. The capstan
is used in moving buildings. Sometimes two or three wheels and axles
are geared together as on a derrick or crane as in Fig. 92.

Fig. 90.?The wheel and axle.Fig. 91.?Windlass used in drawing water from a well.Fig. 92.?A portable crane.Fig. 93.?The wheel and axle considered as a lever.

Fig. 93 is a diagram showing that the wheel and axle acts like a lever. The axis D is the fulcrum, the effort is applied at F, at the extremity of a radius of the wheel[Pg 137][Pg 138] and the resisting weight W at the extremity of a radius of the axle. Hence, if Df, the effort distance, is three times Dw, the weight distance, the weight that can be supported is three times the effort. Here as in the lever,[Pg 139]f ? Df = w ? Dw, or w:f = Df:Dw, or the ratio of the weight to the effort equals the ratio of the radius of the wheel to the radius of the axle.
This is therefore the mechanical advantage of the wheel and axle. Since
the diameters or circumferences are in the same ratio as the radii
these can be used instead of the radii. Sometimes, when increased speed
instead of increased force is desired, the radius of the wheel or part
to which power is applied is less than that of the axle. This is seen in
the bicycle, buzzsaw, and blower. Sometimes geared wheels using the
principle of the wheel and axle are used to reduce speed, as in the transmission of an automobile (see Fig. 94), or the reducing gear of a steam turbine. (See Figs. 95 and 293.)
A bevel gear is frequently used to change the direction of the force. (See Fig. 94.)

Fig. 96.?A single movable pulley.Fig. 97.?Block and tackle.Fig. 98.?The fixed pulley considered as a lever.Fig. 99.?The movable pulley considered as a lever.

125. The Pulley.?The pulley consists of a wheel
turning on an axis in a frame. The wheel is called a sheave and the
frame a block. The rim may be smooth or grooved. The grooved rim is used
to hold a cord or rope. One use of the pulley is to change the direction of the acting force as in Fig. 84, where pulley B changes a horizontal pull at H to a downward force and pulley A changes this into an upward force lifting the weight W. These pulleys are fixed and simply change the direction. Without considering the loss by friction, the pull at W will equal that at F. Sometimes, a pulley is attached to the weight and is lifted with it. It is then called a movable pulley. In Fig. 96 the movable pulley is at P, a fixed pulley is at F. When fixed pulleys
are used, a single cord runs through from the weight to the effort, so
that if a force of 100 lbs. is applied by the effort the same force is
received at the weight. But with movable pulleys several sections of
cord may extend upward from the weight each with the force of the effort
upon it. By this arrangement, a weight several times larger than the
effort can be lifted. Fig. 97 represents[Pg 140] what is called a block and tackle. If a force of 50 lbs. is exerted at F, each section of the rope will have the same tension and hence the six sections of the rope will support 300 lbs. weight. The mechanical advantage of the pulley or the ratio of the weight to the effort, therefore, equals the number of sections of cord supporting the weight.
The fixed pulley represents a lever, see Fig. 98, where the effort and
weight are equal. In the movable pulley, the fulcrum (see Fig. 99) is at
D; the weight, W, is applied at the center of the pulley and the effort at F. The weight [Pg 141]distance, Dw, is the radius, and the effort distance, Df, is the diameter of the pulley. Since W/F = Df / Dw = 2 in a movable pulley, the weight is twice the effort, or its mechanical advantage is 2.

Important Topics

1. Wheel and Axle, Law of Wheel and Axle.
2. Pulley, Fixed and Movable, Block and Tackle, Law of Pulley.

Exercises

1. Why do door knobs make it easier to unlatch doors? What simple machine do they represent? Explain.
2. What combination of pulleys will enable a 160-lb. man to raise a 900-lb. piano?
3. When you pull a nail with an ordinary claw hammer, what is the effort arm? the resistance arm?
4. How much work is done by the machine in problem 2 in lifting the
piano 20 ft.? How much work must be done upon the machine to do this
work?
5. The pilot wheel of a boat has a diameter of 60 in.; the diameter
of the axle is 6 in. If the resistance is 175 lbs., what force must be
applied to the wheel?
6. Four men raise an anchor weighing {1 1/2} tons, with a capstan
(see Fig. 110) having a barrel 9 in. in diameter. The circle described
by the hand-spikes is {13 1/2} ft. in diameter. How much force must each
man exert?

Fig. 100.?The Capstan.

7. A bicycle has a 28-in. wheel. The rear sprocket is 3 in. in diameter,[H]
the radius of the pedal crank is 7 in.; 24 lbs. applied to the pedal
gives what force on the rim of the wheel? What will be the speed of the
rim when the pedal makes one revolution a second?
8. Measure the diameters of the large and small pulleys on the sewing-machine at your home. What mechanical advantage[Pg 142] in number of revolutions does it give? Verify your computation by turning the wheel and counting the revolutions.
9. What force is required with a single fixed pulley to raise a
weight of 200 lbs.? How far will the effort move in raising the weight
10 ft.? What is the mechanical advantage?
10. In the above problem substitute a single movable pulley for the fixed pulley and answer the same questions.
11. What is the smallest number of pulleys required to lift a
weight of 600 lbs. with a force of 120 lbs.? How should they be
arranged?
12. A derrick in lifting a safe weighing 2 tons uses a system of
pulleys employing 3 sections of rope. What is the force required?
13. Name three instances where pulleys are used to do work that otherwise would be difficult to do.
14. Draw a diagram for a set of pulleys by means of which 100 lbs. can lift 400 lbs.

(5) The Inclined Plane. Efficiency

126. Efficiency.?The general law of machines which states
that the work done by a machine equals the work put into it requires a
modification, when we apply the law in a practical way, for the reason
that in using any machine there is developed more or less friction due
to parts of the machine rubbing on each other and to the resistance of
the air as the parts move through it. Hence the statement of the law
that accords with actual working conditions runs somewhat as follows: The work put into a machine equals the useful work done by the machine plus the wasted work done by it. The efficiency of a machine is the ratio of the useful work done by it to the total
work done on the machine. If there were no friction or wasted work, the
efficiency would be perfect, or, as it is usually expressed, would be
100 per cent. Consider a system of pulleys into which are put 600
ft.-lbs. of work. With 450 ft.-lbs. of useful work resulting, the
efficiency would be 450 ? 600 = {3/4}, or 75 per cent. In this case 25
per cent. of the[Pg 143]
work done on the machine is wasted. In a simple lever the friction is
slight so that nearly 100 per cent. efficiency is often secured.
Some forms of the wheel and axle have high efficiencies as in
bicycles with gear wheels. Other forms in which ropes are employed have
more friction. Pulleys have sometimes efficiencies as low as 40 per
cent. when heavy ropes are used.127. Inclined Plane.?We now come to a type of simple machine of lower efficiency
than those previously mentioned. These belong to the inclined plane
group, which includes the inclined plane (see Fig. 101), the wedge and
the screw. They are extensively used, however, notwithstanding their low
efficiency, on account of often giving a high mechanical advantage. The
relation between these machines may be easily shown, as the wedge is obviously a double inclined plane. In Art. 82 it is shown that the effort required to hold a weight upon an inclined plane is to the weight supported as the height of the plane is to its length.

Fig. 101.?An inclined plane.

Or while the weight is being lifted the vertical height BC, the effort has to move the length of the plane AC. Since by the law of machines the effort times its distance equals the weight times its distance, or the weight is to[Pg 144]
the effort as the effort distance is to the weight distance, therefore
the mechanical advantage of the inclined plane is the ratio of the
length to the height of the inclined plane.Inclined planes are used to raise heavy objects short
distances, as barrels into a wagon, and iron safes into a building.
Stairways are inclined planes with steps cut into them.128. The Wedge.?Wedges are used to separate objects, as in
splitting wood (see Fig. 102), cutting wood, and where great force is to
be exerted for short distances. An axe is a wedge, so is a knife. A
fork consists of several round wedges set in a handle. The edge of any
cutting tool is either an inclined plane or a wedge. Our front teeth are
wedges. Numerous examples of inclined planes may be seen about us.
No definite statement as to the mechanical advantage of the wedge
can be given as the work done depends largely on friction. The force
used is generally applied by blows on the thick end. In general, the
longer the wedge for a given thickness the greater the mechanical
advantage.

Fig. 102.?One use of the wedge.

129. The Screw.?The screw is a cylinder around whose
circumference winds a spiral groove. (See Fig. 103.) The raised part
between the two adjacent grooves is the thread of the screw. The screw turns in a block called a nut,
within which is a spiral groove and thread exactly corresponding to
those of the screw. The distance between two consecutive threads
measured parallel to the axis is called the pitch of the screw.
(See Fig. 104.) If the thread winds around the cylinder ten times in the
space of 1 in., the screw is said to have ten threads to the inch, the
pitch being {1/10} in. The screw usually is turned[Pg 145]
by a lever or wheel with the effort applied at the end of the lever, or
at the circumference of the wheel. While the effort moves once about
the circumference of the wheel the weight is pushed forward a distance
equal to the distance between two threads (the pitch of the screw). The
work done by the effort therefore equals F ? 2?r, r being the radius of the wheel, and the work done on the weight equals W ? s, s being the pitch of the screw. By the law of machines F ? 2?r = W ? s or W / F = (2?r) / s. Therefore the mechanical advantage of the screw equals (2?r) / s.
Since the distance the weight moves is small compared to that the power
travels, there is a great gain in force. The screw is usually employed
where great force is to be exerted through small distances as in
the vise (Fig. 105) the jack screw (Fig. 106), screw clamps, to
accurately measure small distances as in the micrometer (Fig. 107) and
spherometer, and to lessen the motion in speed-reducing devices. The
worm gear (Fig. 108) is a modification of the screw that is sometimes
used where a considerable amount of speed reduction is required.

Important Topics

Exercises

1. A plank 12 ft. long is used to roll a barrel weighing 200 lbs.
into a wagon 3 ft. high. Find the force required parallel to the
incline.
2. How long a plank will be needed to roll an iron safe weighing
1-1/2 tons into a wagon 3 ft. high using a pull of 600 lbs. parallel to
the incline.
3. An effort of 50 lbs. acting parallel to the plane prevents a
200-lb. barrel from rolling down an inclined plane. What is the ratio of
the length to the height of the plane?
4. A man can push with a force of 150 lbs. and wishes to raise a
box weighing 1200 lbs. into a cart 3 ft. high. How long a plank must he
use?
5. The radius of the wheel of a letter press is 6 in., the pitch of
its screw is 1/4 in. What pressure is produced by a force of 40 lbs.?
6. The pitch of a screw of a vice is 1/4 in., the handle is 1 ft.
long. what pressure can be expected if the force used is 100 lbs.?
7. A jackscrew is used to raise a weight of 2 tons. The bar of the
jackscrew extends 2 ft. from the center of the screw. There are two
threads to the inch. Find the force required.

(6) Friction, Its Uses and Laws

130. Friction.?Although often inconvenient and expensive,
requiring persistent and elaborate efforts to reduce it to a minimum,
friction has its uses, and advantages. Were it not for friction between
our shoes and the floor or sidewalk, we could not keep our footing. Friction is the resistance that must be overcome when one body moves over another. It is of two kinds, sliding and rolling. If one draws a block and then a car of equal weight along a board, the force employed in each case being measured[Pg 148]
by a spring balance, a large difference in the force required will be
noticed, showing how much less rolling friction is than sliding
friction.131. Ways of Reducing Friction.?(a) Friction is often caused
by the minute projections of one surface sinking into the depressions
of the other surface as one moves over the other. It follows, therefore,
that if these projections could be made as small as possible that
friction would be lessened. Consequently polishing is one of the
best means for reducing friction. In machines all moving surfaces are
made as smooth as possible. In different kinds of materials these little
ridges and depressions are differently arranged. (b) In Fig. 109 the
friction between R and S would be greater than between R and T. In R and S the surfaces will fit closer together than in R and T. The use of different materials will reduce friction.
The iron axles of car wheels revolve in bearings of brass. Jewels are
used in watches for the same reason. (c) Another very common method of
reducing friction is by the use of lubricants. The oil or grease used fills up the irregularities of the bearing surfaces and separates them. Rolling friction
is frequently substituted for sliding friction by the use of ball and
roller bearings. These are used in many machines as in bicycles,
automobiles, sewing machines, etc. (See Fig. 110.)

Fig. 109.?The friction between R and S is greater than between R and T.

132. Value of Friction.?Friction always hinders motion
and whenever one body moves over or through another the energy used in
overcoming the friction is transformed into heat which is taken up by
surrounding bodies and usually lost. Friction is therefore the great
obstacle to[Pg 149] perfect efficiency in machines. Friction, however, like most afflictions has its uses.
We would find it hard to get along without it. Without friction we
could neither walk nor run; no machines could be run by belts; railroad
trains, street cars, in fact all ordinary means of travel would be
impossible, since these depend upon friction between the moving power
and the road for propulsion.

Fig. 110.?Timken roller bearings. As used in the front wheel of an automobile.

133. Coefficient of Friction.?The ratio between the friction when motion is just starting and the force pushing the surfaces together is called the coefficient of friction.
If the block in Fig. 111 is drawn along the board with uniform
motion, the reading of the spring balances indicates the amount of
friction. Suppose the friction is found to be 500 g., and the weight of
the block to be 2000 g.[Pg 150] Then the coefficient of friction for these two substances will be {500/2000} = {1/4}, or 25 per cent.134. Laws of Friction, Law I.?The friction when motion is occurring between two surfaces is proportional to the force holding them together.
Thus if one measures the friction when a brick is drawn along a board,
he will find that it is doubled if a second brick is placed on the
first. On brakes greater pressure causes greater friction. If a rope is
drawn through the hands more pressure makes more friction.

Fig. 111.?A method for testing the friction between surfaces.

Law II.?Friction is independent of the extent of surface in contact.
Thus a brick has the same friction drawn on its side as on its edge,
since, although the surface is increased, the weight is unchanged.Law III.?Friction is greatest at starting, but after starting is practically the same for all speeds.135. Fluid Friction.?When a solid moves through a fluid, as
when a ship moves through the water or railroad trains through the air,
the resistance encountered is not the same as with solids but increases
with the square of the velocity for slow speeds and for high speeds at a
higher rate. This is the reason why it costs so much to increase the
speed of a fast train, since the resistance of the air becomes the
prominent factor at high speeds. The resistance to the motion of a ship
at high speed is usually[Pg 151]
considered to increase as the cube of the velocity so that to double
the speed of a boat its driving force must be eight times as great.

Important Topics

Exercises

1. How long must an inclined plane be which is 10 meters high to
enable a car weighing 2000 kg. to be pushed up its length by a force of
100 kg. parallel to the incline?
2. State how and where friction is of use in the operation of the inclined plane, the wedge, the screw, the wheel and axle.
3. A wheelbarrow has handles 6 ft. long. If a load of 300 lbs. is
placed 18 in. from the axis of the wheel, what force placed at the end
of the handles will be required to lift it?
4. A jackscrew has 3 threads to the inch, and the lever used to
turn it is 4 ft. long. If the efficiency of the screw is 60 per cent.,
what force must be applied to raise a load of 5 tons?
5. In problem 4 how far must the force move in raising the weight 3
in. Compute the work done upon the weight, the work done by the power
and the efficiency of the machine from these two amounts of work.
6. What simple machines are represented in a jackknife, a sewing-machine, a screw-driver, a plane, a saw, a table fork?
7. A laborer carries 1500 lbs. of brick to a platform 40 ft. high. How much useful work does he do?
8. If he weighs 150 lbs. and his hod weighs 10 lbs., how much
useless work does he do in taking 30 trips to carry up the bricks of
problem 7? What is his efficiency?
9. If the laborer hoists the brick of problem 7 in a bucket
weighing 50 lbs., using a fixed pulley and rope, what is the useless
work done if it takes 12 trips to carry up the brick? What is the
efficiency of the device?
10. The efficiency of a set of pulleys is 70 per cent. How much
force should be applied if acting through 100 ft. it is to raise a load
of 400 lbs. 20 ft.?[Pg 152]
11. The spokes of the pilot wheel of a motor-boat are 1 ft. long,
the axle around which the rudder ropes are wound is 3 in. in diameter.
What effort must be applied if the tension in the ropes is 50 lbs.?
12. Why are the elevated railway stations frequently placed at the
top of an incline, the tracks sloping gently away in both directions?
13. The screw of a press has 4 threads to the inch and is worked by
a lever of such length that an effort of 25 lbs. produces a force of 2
tons. What is the length of the lever?
14. It takes a horizontal force of 10 lbs. to draw a sled weighing
50 lbs. along a horizontal surface. What is the coefficient of friction?
15. The coefficient of rolling friction of a railroad train on a
track is 0.009. What pull would an engine have to exert to haul a train
weighing 1000 tons along a level track?
16. How heavy a cake of ice can be dragged over a floor by a
horizontal force of 20 lbs., if the coefficient of friction is 0.06?
17. The coefficient of friction of iron on iron is 0.2. What force can a switch engine weighing 20 tons exert before slipping?
18. Using a system of pulleys with a double movable block a man
weighing 200 lbs. is just able to lift 600 lbs. What is the efficiency
of the system?
19. What is the horse-power of a pump that can pump out a cellar full of water 40 ft. ? 20 ft. by 10 ft. deep, in 30 minutes?
20. How many tons of coal can a 5 horse-power hoisting engine raise
in 30 minutes from a barge to the coal pockets, a height of 50 ft.?

(7) Water Power

136. Energy of Falling Water.?The energy of falling and
running water has been used from the earliest times for developing power
and running machinery. The energy is derived from the action of the
moving water in striking and turning some form of water-wheel, several varieties of which are described below.The Overshot Wheel.?The overshot wheel (Fig. 112) is turned
by the weight of the water in the buckets. It was formerly much used in
the hilly and mountainous[Pg 153]
sections of this country for running sawmills and grist mills as it is
very easily made and requires only a small amount of water. Its
efficiency is high, being from 80 to 90 per cent., the loss being due to
friction and spilling of water from the buckets. To secure this high
efficiency the overshot wheel must have a diameter equal to the height
of the fall which may be as much as 80 or 90 ft.

Fig. 112.?Overshot water wheel.

Fig. 113.?Undershot water wheel.Fig. 114.?Diagram illustrating the principle of the Pelton wheel.

The Undershot Wheel.?The old style undershot wheel (Fig.
113) is used in level countries, where there is little fall, often to
raise water for irrigation. Its efficiency is very low, seldom rising
more than 25 per cent. The principle of the undershot wheel, however, is
extensively used in the water motor and the Pelton wheel (Fig. 114). In
these the water is delivered from a nozzle in a jet against the lower
buckets of the wheel. They have an efficiency of about 80 per cent. and
are much used in cities for running small machines, washing machines,
pipe[Pg 154] organ blowers, etc., and in mountainous districts where the head is great.

Fig. 115.?Diagram of a hydro-electric power house showing a vertical turbine A with penstock B and tail race C.

Fig. 116.?The outer case of a turbine showing the mechanism for controlling the gates.Fig. 117.?Inner case of a turbine showing the gates and the lower end of the runner within.

Fig. 118.?The runner of a turbine.Fig. 119.?Turbine and generator of the Tacoma hydro-electric power plant.

137. The Turbine.?The turbine is now used more than any
other form of water-wheel. It was invented in 1827 by De Fourneyron in
France. It can be used with a small or large amount of water, the power
depending on the head (the height of the water, in the reservoir above
the wheel). It is the most efficient type of water-wheel, efficiencies
of 90 per cent. often being obtained. The wheel is entirely under water
(Fig. 115). It is enclosed in an outer case (Fig. 116) which is
connected with the reservoir by a penstock or pipe and is always kept
full of water. The wheel itself is made in two parts, a rotating part
called the runner (see Fig. 118) and an inner case[Pg 155][Pg 156]
(Fig. 117) with gates that regulate the amount of water entering the
wheel. This case has blades curved so that the water can strike the
curved blades of the rotating part (Fig. 118) at the angle that is best
adapted to use the energy of the water. The water then drops through the
central opening into the tail race below (see Fig. 115).[Pg 157]
The energy available is the product of the weight of the water and the
head. The turbine is extensively used to furnish power for generating
electricity at places where there is a sufficient fall of water. The
electrical energy thus developed is transmitted from 50 to 200 miles to
cities where it is used in running street cars, electric lighting, etc.
Turbines can be made to revolve about either vertical or horizontal
axes. Fig. 119 represents a horizontal water turbine connected to a dynamo. Compare this with the vertical turbine in Fig. 115.

Exercises

1. Does a person do more work when he goes up a flight of stairs in 5 seconds than when he goes up in 15 seconds? Explain.
2. A motorcycle has a 4 horse-power motor and can go at a rate of 50 miles per hour. Why cannot 4 horses draw it as fast?
3. What is the efficiency of a motor that is running fast but doing no useful work?
4. What horse-power can be had from a waterfall, 12 ft. high, if 20 cu. ft. of water pass over it each second?
5. What is the horse-power of a fire engine if it can throw 600 gallons of water a minute to a height of 100 ft.?
6. Why are undershot wheels less efficient than the overshot wheel or turbine?
7. A revolving electric fan is placed on the stern of a boat. Does
the boat move? Why? Place the fan under water. Does the boat now move?
Why?
8. Why does an electric fan produce a breeze?
9. Explain the action of the bellows in an organ.
10. At Niagara Falls the turbines are 136 ft. below the surface of
the river. Their average horse-power is 5000 each. 430 cu. ft. of water
each second pass through each turbine. Find the efficiency.
11. At Laxey on the Isle of Man is the largest overshot wheel now
in use. It has a horse-power of 150, a diameter of 72.5 ft., a width of
10 ft., and an efficiency of 85 per cent. How many cubic feet of water
pass over it each second?[Pg 158]
12. The power plant at the Pikes Peak Hydro-electric Company
utilizes a head of 2150 ft., which is equal to a pressure of 935 lbs.
per square inch, to run a Pelton wheel. If the area of the nozzle is 1
sq. in. and the jet has a velocity of 22,300 ft. per minute, what is the
horse-power developed if the efficiency is 80 per cent.?
13. A test made in 1909 of the turbines at the Centerville power
house of the California Gas and Electric Corporation showed a maximum
horse-power of 9700, speed 400 r.p.m. under a head of 550 ft. The
efficiency was 86.25 per cent. How many cubic feet of water passed
through the turbines each second?
14. The turbine in the City of Tacoma Power Plant (see Fig. 120)
uses a head of 415 ft. 145 cu. ft. a second pass through the turbine.
Calculate the horse-power.
15. In problem 14, what is the water pressure per square inch at the turbine?
16. The power plant mentioned in problem 13 develops 6000 kw. What is the efficiency?